ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 1, pp. 57–63.
Pleiades Publishing, Inc., 2011.
Original Russian Text
G.Sh. Tsitsiashvili, M.A. Osipova, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 1, pp. 66–73.
COMMUNICATION NETWORK THEORY
A Zero-or-One Law in Aggregated
Closed Queueing Networks
G. Sh. Tsitsiashvili and M. A. Osipova
Institute of Applied Mathematics, Far East Branch
of the Russian Academy of Sciences, Vladivostok
Received December 18, 2009; in ﬁnal form, October 12, 2010
Abstract—For a closed queueing network with single-server nodes, we prove that if the total
number of requests, the number of servers in one of the nodes, and service rates in all other
nodes are made n times as large, then the stationary number of requests in the multiserver
node divided by n converges in probability as n →∞to a positive constant, determined by
parameters of the original network, with geometric convergence rate. Single-server nodes in
the constructed network can be interpreted as repair nodes, the multiserver node as a set of
workplaces, and requests as elements in a redundancy-with-repair model.
Exponential closed queueing networks ﬁnd numerous applications in modeling communication
systems, data transmission and data processing systems, etc. [1–5]. Asymptotic relations for such
networks is a tool to simplify computations in estimation of normalizing constants for product-form
It is known that if in an exponential closed queueing network with single-server nodes the
number of requests and service rates are made n times as large, then, as n →∞, the closed network
transforms into an open one where the most loaded node of the original network becomes a source of
incoming requests and a drain for outgoing requests [1, p. 47, equation (3.3.1); 6; 7]. However, this
result cannot be applied to redundancy-with-repair models , since a node containing workplaces
could be excluded from consideration.
In the present paper, in one of the nodes of a closed network, instead of multiplying the service
rate by n we introduce n servers with the original rate (by analogy with [9,10]). In a network thus
constructed, single-server nodes can be interpreted as repair nodes, the multiserver node as a set of
workplaces, and requests as elements which fail at workplaces and are reconstructed at repair nodes
in a redundancy-with-repair model. Another interpretation is an assumption that the multiserver
node consists of a set of consumers, while requests are products consumed at the workplace node
and regenerated at single-server nodes.
For this model we prove that the ratio of the stationary number of requests in a multiserver
node to the number of servers converges as n →∞to a constant b>0, determined by parameters
of the original network, with geometric convergence rate. This statement makes it possible, ﬁrst,
to obtain a zero-and-one law for the probability that all servers at the multiserver node are busy
Supported in part by the Far East Branch of the Russian Academy of Sciences, project nos. 09-1-P2-07